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Argumentation-Based Defeasible Reasoning For

Existential Rules

Abdelraouf Hecham, Madalina Croitoru, Pierre Bisquert

To cite this version:

Abdelraouf Hecham, Madalina Croitoru, Pierre Bisquert. Argumentation-Based Defeasible Reasoning

For Existential Rules. AAMAS: Autonomous Agents and MultiAgent Systems, May 2017, Sao Paulo,

Brazil. pp.1568-1569. �lirmm-01520171�

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Argumentation-Based Defeasible Reasoning For Existential Rules

Abdelraouf Hecham

1

, Pierre Bisquert

2

, Madalina Croitoru

1

1GraphIK INRIA, University of Montpellier

2GraphIK INRIA, French Institute of Research in Agronomy (INRA)

Abstract

Logic based argumentation allows for defeasible reasoning over monotonic logics. In this paper, we introduce DEFT, a tool implementing argumentative defeasible reasoning over existential rules. We explain how DEFT overcomes deriva-tion loss and discuss DEFT’s empirical behavior.

Research Context

We are interested in argumentation based defeasible rea-soning (Dung 1995; Prakken and Sartor 1997; Garc´ıa and Simari 2004) for existential rules, a first order logic sub-set employed for sharing, reuse and reasoning over large databases on the Semantic Web (SW) (Cal`ı, Gottlob, and Lukasiewicz 2012) and we focus on the tractable frag-ments of existential rules, Datalog± (Cal`ı, Gottlob, and Lukasiewicz 2012). Defeasible Datalog±was proposed in (Martinez et al. 2014; Deagustini et al. 2015) but it does not account for the possibility of lossof deduced informa-tion that can render the reasoning process non determinis-tic. We improve upon the state of the art by proposing a new hypergraph-based algorithm that will prevent derivation loss.

The contribution of the paper is to provide the first ar-gumentation based toolfor defeasible Datalog±in the lit-erature: DEFT (Defeasible Datalog±Tool) and compare its behavior with respect to argumentation tools in the litera-ture relevant for defeasible reasoning with ambiguity prop-agation: ASPIC+ (Prakken 2010) with its grounded seman-tics (Governatori et al. 2004) and DeLP (Garc´ıa and Simari 2004) with its dialectical trees.

Chase Derivation Loss

Defeasible Datalog± extends Datalog±to include defea-sible facts and defeadefea-sible rules. Datalog± refers to the decidable fragment of existential rules (Cal`ı, Gottlob, and Lukasiewicz 2012) that extends the Datalog language with existential variablesin the rule head. Defeasible reasoning in this context is based on dialectical trees, which, in turn, are based on the classical notion of derivation (i.e. the suc-cessive application of a set of rules on a set of facts). The rule application mechanism is called the chase.The chase pro-cedure is equipped with a restriction test, called derivation

reducer, for detecting when the rule application becomes re-dundant. In the literature several chases are studied (Baget et al. 2014). Here we only restrict ourselves to the derivation reducer used for the Restricted chase (Fagin et al. 2005). Using a derivation reducer may induce a loss of rule appli-cations depending on the order in which rules are applied. While the order of applications does not impact the final model of the saturated knowledge base (and thus entailment in Datalog±), it does affect the set of extracted deriva-tions. This is important since dialectical-tree-based defeasi-ble reasoning relies on the set of extracted derivations.

To allow for lossless derivation extraction we use an adap-ted combinatorial structure called Graph of Atom Depen-dency (GAD). This structure can be seen as an improvement over the chase graph (Cal`ı, Gottlob, and Lukasiewicz 2012) that allows keeping track of all the generated atoms during the chase procedure. In the GAD the nodes correspond to facts and the labelled directed hyperedges to the rule appli-cations. The intuition behind the use of the GAD is that, for a given GAD and a given atom, there is a one-to-one mapping, up to derivation equivalence, between the set of hyperpaths to an atom f and the set of derivations to f . The problem of obtaining all minimal derivations of f can thus be trans-formed into the problem of generating all minimal hyper-paths of f in the GAD. For every hyperpath of the GAD we can construct a derivation, and for every derivation there ex-ists a hyperpath. This ensures soundness and completeness of all minimal derivations extraction.

Let us present the chase-based implementation of defea-sible Datalog±called “DEFT”. DEFT uses the GAD struc-ture and relies on the Datalog±dedicated inference engine called “GRAAL” (Baget et al. 2015) that accepts a wide va-riety of formats (OWL2, RuleML and the Datalog±format DLGP (Baget et al. 2015)). We run the restricted chase using the GRAAL framework in order to create the GAD.

Results and discussion

By using the GAD structure, DEFT is the first sound and complete Datalog± defeasible reasoning tool in the litera-ture. The Datalog± features impose a dedicated tool due to two main aspects. First, Datalog± allows for existen-tial rules, which neither ASPIC+ nor DeLP can handle since they cannot express existential variables. This can easily be checked by trying to answer the query q(a, a) with the rule

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p(X) → q(X, Y ) and the fact p(a) using online tools like ASPIC argumentation engine demo1 or DeLPclient2. The

query is entailed if the system does not support existential rules (as the Y variable is mapped to all known constants). Second, since Datalog± allows for weak negation but not classic negation in the body of rules, DEFT can simply use a fast chase mechanism, whereas existing argumentation-based reasoning tools rely on resolution-argumentation-based inference mechanisms (Bryant and Krause 2008) since they account for classic negation, which induces a large computational overhead.

We conducted an empirical evaluation of DEFT in order to measure its performance w.r.t to DeLP and ASPIC+ 3.

The experiments are built upon a pre-established defeasi-ble reasoning benchmark proposed in (Maher et al. 2001). The benchmark we consider is composed of 5 parameter-ized knowledge bases (also known as theories): Chain The-ory tests performance when faced with a simple chain of rules; Circle Theory tests infinite loops (cycles), Levels and Trees Theories test a large number of arguments with small derivations and Teams Theory tests performance w.r.t. a sizeable number of conflicts. The results are shown in the table below.

The salient points of this evaluation are two-fold. First, DEFT is the only tool able to reason with the class of existential rules guaranteed to stop in forward chaining (namely, Finite Expansion Set (Baget et al. 2014)). Second, DEFT out-performs existing argumentation-based defea-sible reasoning tool for general logical fragments with only weak negation.

In future work we plan to investigate the semantic relation between defeasible Datalog± and the floating conclusions from the work of (Arioua and Croitoru 2016). We also plan to use DEFT in human reasoning akin frameworks for irra-tionality (Bisquert et al. 2016b; 2016a).

Acknowledgment

This work has been supported by the French National Re-search Agency, ASPIQ project ANR-12-BS02-0003.

References

Arioua, A., and Croitoru, M. 2016. A dialectical proof the-ory for universal acceptance in coherent logic-based argu-mentation frameworks. In ECAI, volume 285, 55–63. IOS Press.

Baget, J.-F.; Garreau, F.; Mugnier, M.-L.; and Rocher, S. 2014. Extending acyclicity notions for existential rules. In ECAI, 39–44.

Baget, J.-F.; Lecl`ere, M.; Mugnier, M.-L.; Rocher, S.; and Sipieter, C. 2015. Graal: A toolkit for query answering with existential rules. In International Symposium on Rules

1http://aspic.cossac.org/

ArgumentationSystem/

2http://lidia.cs.uns.edu.ar/delp_client/ 3

We used DeLP implementation in Tweety1.7 libraries and an author provided implementation for ASPIC+.

Table 1: Execution time in seconds

Theory Size DEFT ASPIC+ DeLP

chain (n ) n = 100 201 0.02 0.16 0.97 n = 400 801 0.04 23.57 31.01 n = 800 1601 0.08 ∞ 254.96 n = 2000 4001 0.28 ∞ T.O. cir cl e (n ) n = 100 201 0.02 ∞ 0.99 n = 400 801 0.04 ∞ 49.88 n = 800 1601 0.06 ∞ T.O. n = 2000 4001 0.25 ∞ T.O. lev el s (n ) n = 25 261 0.02 0.02 169.94 n = 4000 40011 1.09 23.22 T.O. n = 8000 80011 2.08 115.68 T.O. n = 11000 110011 2.83 T.O. T.O. teams (n ) n = 2 232 0.09 0.02 6.93 n = 3 808 0.36 0.13 T.O. n = 4 3196 3.08 10.50 T.O. n = 5 15016 25.99 133.73 T.O. tr ees (n, k ) n = 3, k = 5 311 0.10 0.10 227.38 n = 5, k = 5 7811 2.14 109.36 T.O. n = 6, k = 5 39061 T.O. T.O. T.O.

and Rule Markup Languages for the Semantic Web, 328– 344. Springer.

Bisquert, P.; Croitoru, M.; de Saint-Cyr, F. D.; and Hecham, A. 2016a. Formalizing cognitive acceptance of arguments: Durum wheat selection interdisciplinary study. Mind and Machines (to appear).

Bisquert, P.; Croitoru, M.; de Saint-Cyr, F. D.; and Hecham, A. 2016b. Substantive irrationality in cognitive systems. In Proceedings of ECAI 2016.

Bryant, D., and Krause, P. 2008. A review of current defea-sible reasoning implementations. The Knowledge Engineer-ing Review23(03):227–260.

Cal`ı, A.; Gottlob, G.; and Lukasiewicz, T. 2012. A general datalog-based framework for tractable query answering over ontologies. Web Semantics: Science, Services and Agents on the World Wide Web14:57–83.

Deagustini, C.; Martinez, M.; Falappa, M.; and Simari, G. 2015. On the influence of incoherence in inconsistency-tolerant semantics for datalog±.

Dung, P. M. 1995. On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artificial intelligence 77(2):321–357.

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Data exchange: semantics and query answering. Theoretical Computer Science336(1):89–124.

Garc´ıa, A. J., and Simari, G. R. 2004. Defeasible logic pro-gramming: An argumentative approach. Theory and prac-tice of logic programming4:95–138.

Governatori, G.; Maher, M. J.; Antoniou, G.; and Billington, D. 2004. Argumentation semantics for defeasible logic. Journal of Logic and Computation14(5):675–702.

Maher, M. J.; Rock, A.; Antoniou, G.; Billington, D.; and Miller, T. 2001. Efficient defeasible reasoning sys-tems. International Journal on Artificial Intelligence Tools 10(04):483–501.

Martinez, M. V.; Deagustini, C. A. D.; Falappa, M. A.; and Simari, G. R. 2014. Inconsistency-tolerant reasoning in datalog± ontologies via an argumentative semantics. In IB-ERAMIA 2014. Springer. 15–27.

Prakken, H., and Sartor, G. 1997. Argument-based extended logic programming with defeasible priorities. Journal of ap-plied non-classical logics7(1-2):25–75.

Prakken, H. 2010. An abstract framework for argumenta-tion with structured arguments. Argument and Computaargumenta-tion 1(2):93–124.

Figure

Table 1: Execution time in seconds

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